Consistency of invariance-based randomization tests
Consistency of invariance-based randomization tests
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DOI：
10.1214/22-aos2200
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发表时间：
2021-04
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影响因子：
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通讯作者：
Edgar Dobriban
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文献类型：
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作者：
Edgar Dobriban
基于不变性的随机化检验（例如排列检验、旋转检验或符号变化）是一类重要且广泛使用的统计方法。它们允许在数据分布的弱假设下进行推断。大多数工作都集中在它们的 I 类错误控制属性上，而它们的一致性属性却很少被理解。我们开发了一个通用框架和一组关于信号加噪声模型中基于不变性的随机化测试的一致性的结果。我们的框架基于表示论的深层数学领域。我们允许变换是一般紧凑拓扑群，例如旋转群，通过一般线性群表示起作用。我们研究具有广义子可加性属性的检验统计量。我们将我们的框架应用于统计学中的许多基本且非常重要的问题，包括稀疏向量检测、噪声中低秩矩阵的测试、线性回归中的稀疏检测以及两个样本测试。与极小极大下限相比，我们可能令人惊讶地发现，在某些情况下，随机化测试以极小极大最佳速率检测信号。
Invariance-based randomization tests -- such as permutation tests, rotation tests, or sign changes -- are an important and widely used class of statistical methods. They allow drawing inferences under weak assumptions on the data distribution. Most work focuses on their type I error control properties, while their consistency properties are much less understood. We develop a general framework and a set of results on the consistency of invariance-based randomization tests in signal-plus-noise models. Our framework is grounded in the deep mathematical area of representation theory. We allow the transforms to be general compact topological groups, such as rotation groups, acting by general linear group representations. We study test statistics with a generalized sub-additivity property. We apply our framework to a number of fundamental and highly important problems in statistics, including sparse vector detection, testing for low-rank matrices in noise, sparse detection in linear regression, and two-sample testing. Comparing with minimax lower bounds, we find perhaps surprisingly that in some cases, randomization tests detect signals at the minimax optimal rate.